Splitting field is countable

Question

I'm trying to prove that if $K$ is a countable field then there exists a countable field $L$ containing $K$ such that every polynomial in $K[X]$ splits in $L$. I know that if $L$ is the splitting field for one such polynomial then $[L:K]$ is finite and so $L$ is countable. I know that $K[X]$ is countable if $K$ is countable. But I can't see why the field containing the roots of all polynomials over $K$ should be countable. Thanks.

Answer 1

HINT: Since $K$ is countable, order the polynomials in $K$ as $p_i$ (for $i\in\mathbb{N}$) and consider a tower of extensions $$K\subset L_0\subset L_1\subset ...$$ such that $p_i$ splits in $L_i$. Then:

• How big is each individual $L_i$?

• How big is the union $L=\bigcup L_i$?

• What is this $L$?